,
,
,
,
or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
,
isn't even rigorously defined in this expression. What exactly do we mean by 
.
,
does not exist in the first place, although you will see why in a calculus course. So the point is that 
?
indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that 
,
or ![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](http://latex.artofproblemsolving.com/2/6/0/260f97a1cf1ea8722ff97a29b83bac7bfe83e577.png)
etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function 
given by the mapping 
is undefined for 
.
is undefined because dividing by ![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](http://latex.artofproblemsolving.com/9/9/3/993c78b62ec2c84b5d77daa8e9a8cd6f70fcf904.png)
So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let 
for each 
which means that via this order topology each subset has an infimum and supremum and 
is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In ![\begin{align*}
a + \infty = \infty + a & = \infty, & a & \neq -\infty \\
a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
\frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
\frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
\frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0).
\end{align*}](http://latex.artofproblemsolving.com/6/d/8/6d8fc1eca5c791d430a7168cd3b37a02104fa318.png)
So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,![\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]](http://latex.artofproblemsolving.com/a/0/b/a0b67076efcc014c78b2023be1151c84b57c1503.png)
So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define 
as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by ![\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]](http://latex.artofproblemsolving.com/8/4/e/84ed3d6ab237df970333b87beaef5311caa46185.png)
which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, 
means complex infinity, since we are in the complex plane now. Here's the catch: division by ![\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]](http://latex.artofproblemsolving.com/1/6/0/160cb939707708ff4e0930724df2928af11d7fd1.png)
where 
and 
Furthermore, we actually have some nice properties with multiplication that we didn't have before. In 
it holds that![\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]](http://latex.artofproblemsolving.com/8/e/0/8e035b4841d20c5ad671c46355cc667ca01e533b.png)
but 
and 
are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with ![\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]](http://latex.artofproblemsolving.com/c/0/4/c044ff37849acad3b8b280b9753539225a1276ca.png)
They behave in a similar way to the Riemann Sphere, with division by 
.
is rounded to the nearest integer and then divided by 
,
Y by OronSH, tacocatz
Multiple states have announced their top students who will advance to the 2025 MATHCOUNTS National Competition in May:
• From Alabama: Henry Gladden of Mobile, Austin Lu of Birmingham, Jessie Shi of Vestavia, and Minlu Wang-He of Auburn.
• From Connecticut: Hayden Hughes of Newtown, Ethan Shi of Riverside, Alex Svoronos of Greenwich and Elaine Zhou of Hamden.
• From Kansas: Haidan Anderson & Jayden Xue of Overland Park, Ruby Jiang of Lawrence, and Christopher Spencer of Manhattan.
• From Massachusetts: Eric Huang of Acton, Shlok Mukund & Brandon Ni of Lexington, and Soham Samanta of Medford.
• From Rhode Island: Kahlan Anderson of the Wheeler School, Julian Bernhoft & Colin Hegstrom of Providence, and Theodora Watson of Barrington.
• From South Carolina: Yukai Hu of Elgin, Justin Peng of Clemson, Geonhoo Shim of Columbia, and Aaron Wang of Mount Pleasant.
===
Updated on 3/11/2025:
• From Hawaii: Isaac Qian, Taehwan Jeon, Hilohak Kwak, and Thien Tran
• From Michigan: Arnav Vunnam, Eric Jin, Akshaj Malraj, and Chaithanya Budida
• From Missouri: Jay Zhou, Charles Yong, Kevin Shi, and Lucas Lai
• From Nevada: Aaron Lei, Maxwell Tsai, Leeoz Nebat, and Solomon Dumont
• From Oregon: Sophia Han, Kevin Cheng, Garud Shah, and Ryan Zhang
• From Texas: Shaheem Samsudeen, Ayush Narayan, Nathan Liu, and James Stewart
• From Wisconsin: Augie Reeder, Junhao Feng, Jiyan Singh, and Lucy Chien
• From Alabama: Henry Gladden of Mobile, Austin Lu of Birmingham, Jessie Shi of Vestavia, and Minlu Wang-He of Auburn.
• From Connecticut: Hayden Hughes of Newtown, Ethan Shi of Riverside, Alex Svoronos of Greenwich and Elaine Zhou of Hamden.
• From Kansas: Haidan Anderson & Jayden Xue of Overland Park, Ruby Jiang of Lawrence, and Christopher Spencer of Manhattan.
• From Massachusetts: Eric Huang of Acton, Shlok Mukund & Brandon Ni of Lexington, and Soham Samanta of Medford.
• From Rhode Island: Kahlan Anderson of the Wheeler School, Julian Bernhoft & Colin Hegstrom of Providence, and Theodora Watson of Barrington.
• From South Carolina: Yukai Hu of Elgin, Justin Peng of Clemson, Geonhoo Shim of Columbia, and Aaron Wang of Mount Pleasant.
===
Updated on 3/11/2025:
• From Hawaii: Isaac Qian, Taehwan Jeon, Hilohak Kwak, and Thien Tran
• From Michigan: Arnav Vunnam, Eric Jin, Akshaj Malraj, and Chaithanya Budida
• From Missouri: Jay Zhou, Charles Yong, Kevin Shi, and Lucas Lai
• From Nevada: Aaron Lei, Maxwell Tsai, Leeoz Nebat, and Solomon Dumont
• From Oregon: Sophia Han, Kevin Cheng, Garud Shah, and Ryan Zhang
• From Texas: Shaheem Samsudeen, Ayush Narayan, Nathan Liu, and James Stewart
• From Wisconsin: Augie Reeder, Junhao Feng, Jiyan Singh, and Lucy Chien
This post has been edited 9 times. Last edited by Chatelet1, Yesterday at 7:27 PM
Reason: TX
Reason: TX
